One simple solution is that both sides play with the same ball that is provided either by the NFL or the home team.
Imagine if each team brought their own soccer ball to a match and the balls were swapped out each time there was a turnover of possession?
Another solution is that the pressure of the ball is checked by a match official at each turnover. There is enough stop-start in football as it is a quick pressure test before the ball goes onto the field isn't going to cause any delay.
“The fundamental cause of the trouble is that in the modern world the stupid are cocksure while the intelligent are full of doubt.” - Bertrand Russell
"When the power of love overcomes the love of power, the world will know peace."
Jimi Hendrix
